Layer potential theory for the anisotropic Stokes system with variable L∞ symmetrically elliptic tensor coefficient

The aim of this paper is to develop a layer potential theory in L2‐based weighted Sobolev spaces on Lipschitz bounded and exterior domains of ℝn , n ≥ 3, for the anisotropic Stokes system with L∞ viscosity tensor coefficient satisfying an ellipticity condition for symmetric matrices with zero matrix trace. To do this, we explore equivalent mixed variational formulations and prove the well‐posedness of some transmission problems for the anisotropic Stokes system in Lipschitz domains of ℝn , with the given data in L2‐based weighted Sobolev spaces. These results are used to define the volume (Newtonian) and layer potentials and to obtain their properties. Then, we analyze the well‐posedness of the exterior Dirichlet and Neumann problems for the anisotropic Stokes system with L∞ symmetrically elliptic tensor coefficient by representing their solutions in terms of the obtained volume and layer potentials.

Choi and Lee 11 have studied the Dirichlet problem for the stationary Stokes system with irregular coefficients. They have proved the unique solvability of the problem in Sobolev spaces on a Lipschitz domain in R n , n ≥ 3, with a small Lipschitz constant, by assuming that the coefficients have vanishing mean oscillations (VMO) with respect to all variables. Existence and pointwise bounds of the fundamental solution for the stationary Stokes system with measurable coefficients in R n (n ≥ 3) have been obtained by Choi and Yang 12 under the assumption of local Hölder continuity of weak solutions of the Stokes system. They also discussed the existence and pointwise bounds of the Green function for the Stokes system with measurable coefficients on unbounded domains where the divergence equation is solvable, particularly on the half-space. The solvability in Sobolev spaces of the conormal derivative problem for the stationary Stokes system with nonsmooth coefficients on bounded Reifenberg flat domains have been proved by Choi et al. 13 (see also Choi et al. 14 ).
The methods of layer potential theory play also a significant role in the study of elliptic boundary value problems with variable coefficients. Mitrea and Taylor 15 have obtained well-posedness results for the Dirichlet problem for the smooth coefficient Stokes system in L p spaces on arbitrary Lipschitz domains in a compact Riemannian manifold and extended the well-posedness results by Fabes et al. 7 from the Euclidean setting to the compact Riemannian setting. Dindos and Mitrea 3 have used the mapping properties of Stokes layer potentials in Sobolev and Besov spaces to show well-posedness results for Poisson problems for the smooth coefficient Stokes and Navier-Stokes systems with Dirichlet boundary condition on C 1 and Lipschitz domains in compact Riemannian manifolds. Well-posedness results for transmission problems for the smooth coefficient Navier-Stokes and Darcy-Forchheimer-Brinkman systems in Lipschitz domains on compact Riemannian manifolds have been obtained by Kohr et al. 16 An alternative approach was employed by Chkadua et al., [17][18][19][20][21][22] where various boundary value problems for variable-coefficient elliptic partial differential equations were reduced to explicit parametrix-based boundary-domain integral equations (BDIEs). Equivalence of BDIEs to the boundary value problems and invertibility of BDIE operators in L 2 and L p -based Sobolev spaces have been analyzed in these papers. Localized BDIEs based on a harmonic parametrix for divergence-form elliptic PDEs with variable matrix coefficients have been also developed; see Chkadua et al. 23 and the references therein.
Amrouche et al. 24 used a variational approach in the analysis of the exterior Dirichlet and Neumann problems for the n-dimensional Laplace operator in weighted Sobolev spaces. Mazzucato and Nistor 25 obtained well-posedness and regularity results for the elasticity equations with mixed conditions on polyhedral domains. Hofmann et al. 26 considered layer potentials in L p spaces for elliptic operators of the form L = −div(A∇u) that act in the upper half-space R n+1 + ∶= {(x, t) ∶ x ∈ R n , t ∈ R + }, n ≥ 2, or in more general Lipschitz graph domains, where A is an (n + 1) × (n + 1) type matrix of L ∞ complex, t-independent coefficients satisfying a uniform ellipticity condition, and solutions of the equation Lu = 0 satisfying De Giorgi-Nash-Moser-type interior estimates. They developed a Calderón-Zygmund-type theory associated with the layer potentials and obtained well-posedness results for related boundary problems in L p and endpoint spaces. Brewster et al. 27 have used a variational approach to obtain well-posedness results for Dirichlet, Neumann, and mixed boundary problems for higher order divergence-form elliptic equations with L ∞ coefficients in locally ( , )-domains and in Besov and Bessel potential spaces (see also Haller-Dintelmann et al. 28 ). Barton 29 has used the Lax-Milgram lemma to construct layer potentials for strongly elliptic differential operators in Banach spaces and generalized many properties of layer potentials for the harmonic equation. Barton and Mayboroda 30 developed layer potentials for second-order divergence elliptic operators with bounded measurable coefficients that are independent of the (n + 1)st coordinate and well-posedness results for related boundary problems with data in Besov spaces.
Girault and Sequeira 31 obtained well-posedness of the exterior Dirichlet problem for the constant coefficient Stokes system in weighted Sobolev spaces on exterior Lipschitz domains in R n for n ∈ {2, 3}, by applying a mixed variational formulation. Angot 32 analyzed some Stokes/Brinkman transmission problems with a scalar viscosity coefficient on bounded domains. Sayas and Selgas 33 developed a variational approach for the constant-coefficient Stokes layer potentials on Lipschitz boundaries, by using the technique of Nédélec. 34 The book by Sayas et al. 35 gives a comprehensive presentation of the basic variational theory for elliptic PDEs in Lipschitz domains. Bȃcuţȃ et al. 36 developed a variational approach for the constant-coefficient Brinkman single-layer potential and used it to analyze the corresponding time dependent exterior Dirichlet problem in R n , n = 2, 3. Alliot and Amrouche 37 have used a variational approach to obtain weak solutions for the exterior Stokes problem in weighted Sobolev spaces (see also Amrouche and Nguyen 38 ).
Kohr et al. 39 obtained the well-posedness results for the isotropic Stokes system with a nonsmooth scalar viscosity coefficient ∈ L ∞ (R 3 ) (see also previous studies [40][41][42] for the Stokes and Navier-Stokes systems with nonsmooth coefficients in compact Riemannian manifolds). Kohr et al. 43 also analyzed transmission problems in weighted Sobolev spaces for anisotropic Stokes and Navier-Stokes systems with an L ∞ strongly elliptic coefficient tensor, in the pseudostress setting.
In this paper, we proceed with the study of transmission and exterior boundary value problems for the anisotropic Stokes system. However, unlike paper, 43 we consider the L ∞ viscosity coefficient tensor satisfying an ellipticity condition only with respect to all symmetric matrices with zero matrix trace (see 1.4). Our purpose is to develop a generalized layer and volume potential theory in L 2 -based weighted Sobolev spaces for such Stokes systems, which does not involve fundamental solutions and hence can be used when the fundamental solutions are not available. To do this, we explore equivalent mixed variational formulations and prove the well-posedness of some transmission problems for the anisotropic Stokes system in Lipschitz domains of R n , with the given data in L 2 -based weighted Sobolev spaces. These results are used to define the volume and layer potentials in terms of solutions of the transmission problems and to obtain the potential properties, without introducing classical explicit integral potential operators. However, when the explicit integral representations of the potentials are available, they will coincide with the variational potentials developed here due to the uniqueness of solutions to the corresponding transmission problems.
Then, we analyze well-posedness of the exterior Dirichlet and Neumann problems for the anisotropic Stokes system with L ∞ tensor coefficient satisfying ellipticity condition (1.4) and represent their solutions in terms of the anisotropic Stokes Newtonian and layer potentials. Although these boundary value problems can be analyzed by variational methods directly, without employing the potential formalism, they are provided here as examples on how, using this formalism, one can easily generalize the classical potential approaches, available for constant-coefficient isotropic problems, to the discontinuous-coefficient anisotropic ones. The potential theory developed in this paper can be also useful when new fundamental solutions and potentials based on them become available. In this case, the potential properties can be obtained from the results developed here.
This paper deals with the potentials in R n , n ≥ 3. Its results can be extended to R 2 as well, but then, the analysis should be done in slightly different weighted Sobolev spaces.
Note that the boundary value problems for the anisotropic Stokes system with L ∞ coefficients considered in this paper can describe physical, engineering, or industrial processes related to the flow of immiscible fluids, or the flow of nonhomogeneous fluids with density dependent viscosity (cf., e.g., Choi et al. 13 ). They appear also in modeling incompressible elastic anisotropic nonhomogeneous/composite materials.

The anisotropic Stokes system with L ∞ symmetrically elliptic tensor coefficient
All along the paper, we use the Einstein summation convention for repeated indices from 1 to n, and the standard notation for the first-order partial derivatives x , = 1, … , n. Let L be a second-order differential operator in the divergence form in an open set Ω ⊆ R n , n ≥ 3, 1≤ , ≤n is the symmetric part of the gradient ∇u. Therefore, the components of the tensor field E(u) are defined by E (u) ∶= 1 2 ( u + u ). The viscosity tensor coefficient A in the operator L consists of n × n matrix-valued functions A = A (x) with essentially bounded, real-valued entries, that is, satisfying the symmetry conditions  (7)). Note that the symmetry conditions (1.3) do not imply the symmetry a i (x) = a i (x), which will be generally not assumed in the paper. We assume that the coefficients satisfy the following relaxed ellipticity condition, which asserts that there exists a constant c A > 0 such that for almost all x ∈ Ω, where | | 2 = i i . Therefore, the ellipticity condition (1.4) is assumed only for all symmetric matrices = ( i ) i, =1, … ,n ∈ R n×n (cf. Oleinik et al. 44 , eqs. (3.1) and (3.2)), having zero matrix trace, ∑ n i=1 ii = 0. In view of (1.2), A is endowed with the norm The symmetry conditions (1.3) allow us to express the operator L in the equivalent forms Note that the first equality in (1.6) has not been encountered in our publication, 43 where the coefficients of the fourth-order tensor A have been assumed to satisfy the strong ellipticity condition similar to the second condition in (1.4) but for all (not only symmetric and zero-trace) matrices (see Kohr et al. 43 , eq. 2 and 3). The more restrictive ellipticity condition in paper 43 allowed to explore there the associated nonsymmetric pseudostress setting. In this paper, we require the symmetry conditions (1.3) and the ellipticity condition (1.4) only for symmetric zero-trace matrices and develop our results in the symmetric stress setting. This approach allows us to obtain properties of layer potentials for the Stokes system with L ∞ variable coefficients generalizing well-known results for constant coefficients.
Let u be an unknown vector field, be an unknown scalar field, and f and g be, respectively, vector and scalar fields defined in Ω ⊆ R n . Then, the equations determine the Stokes system which describes viscous compressible fluid flows with variable anisotropic viscosity tensor coefficient A depending on the physical properties of the fluid, such as, for example, the given fluid temperature. 45,46 If g = 0, then the fluid is incompressible. According to (1.6) and (1.7), the Stokes operator  can be written in any of the equivalent forms Under condition (1.4), the anisotropic Stokes system (1.8) is Agmon-Douglis-Nirenberg elliptic (see Lemma 15).

Isotropic case
For the isotropic case, the viscosity tensor A in (1.2) has the form (cf., e.g., appendix III, part I, section 1 in Temam 47 ) where , ∈ L ∞ (Ω) and with a constant c > 0. Then, for any symmetric matrix = ( i ) 1≤i, ≤n ∈ R n×n such that ii = ∑ n i=1 ii = 0. Therefore, the symmetric ellipticity condition (1.4) is satisfied as well, and hence, our results are also applicable to the Stokes system in the isotropic case. If > 0 is a constant and g = 0, then (1.8) reduces to the well-known isotropic incompressible Stokes system with constant viscosity .
Let Ω + be a bounded Lipschitz domain in R n , that is, an open connected set whose boundary Ω is locally the graph of a Lipschitz function and is connected. We further assume that n ≥ 3 unless explicitly stated otherwise. Sometimes, we will write just Ω instead of Ω + . Let Ω − ∶= R n ∖Ω + be the corresponding exterior Lipschitz domain. LetE ± denote the operator of extension of functions by zero outside Ω±.

L 2 -based Sobolev spaces
Given a Banach space , its topological dual is denoted by  ′ , and the notation ⟨· , ·⟩ X means the duality pairing of two dual spaces defined on a set X ⊆ R n .
Let Ω ′ be a nonempty open set in R n or just R n . Let L 2 (Ω ′ ) denote the Lebesgue space of (equivalence classes of) measurable, square-integrable functions on Ω ′ , and L ∞ (Ω ′ ) denote the space of (equivalence classes of) essentially bounded measurable functions on Ω ′ . Let us define the L 2 -based Sobolev space Here, L 2 (Ω ′ ) n denotes the space of vector-valued functions whose components belong to the scalar space L 2 (Ω ′ ). Similar notations are assumed also for other vector-valued and matrix-valued spaces.
Let (Ω ′ ) ∶= C ∞ 0 (Ω ′ ) denote the space of infinitely differentiable functions with compact support in Ω ′ , equipped with the inductive limit topology. Let  ′ (Ω ′ ) denote the corresponding space of distributions on Ω ′ , that is, the dual of the space (Ω ′ ).
Let Ω ′′ be either a bounded Lipschitz domain or the exterior of a bounded Lipschitz domain in R n . The spaceH 1 (Ω ′′ ) is the closure of (Ω ′′ ) in H 1 (R n ). It can be also characterized as where y is the surface measure on Ω (see, e.g., Proposition 2.5.1 in Mitrea and Wright 5 ). The dual of H s ( Ω) is the space H −s ( Ω), and we set H 0 ( Ω) = L 2 ( Ω). Let H s ( Ω) n denote the space of vector-valued functions whose components belong to H s ( Ω). The dual of H s ( Ω) n is the space H −s ( Ω) n . All L 2 -based Sobolev spaces mentioned above are Hilbert spaces. The following well-known trace theorem holds true (cf. Costabel

Weighted Sobolev spaces
Let |x| = (x 2 1 + … + x 2 n ) 1 2 denote the Euclidean distance of a point x = (x 1 , … , x n ) ∈ R n to the origin of R n . Let be the weight function (2.4)

Weighted Sobolev spaces on R n
The weighted Lebesgue space L 2 ( −1 ; R n ) is defined by and has a Hilbert space structure with respect to the inner product and the associated norm We also consider the weighted Sobolev space (cf. Definition 1.1 in Alliot and Amrouche 37 and Theorem I.1 in Hanouzet 51 ), which is also a Hilbert space with the norm defined by  33 in the case n = 3), and thus, the dual This seminorm is a norm on the space  1 (R n ) and is equivalent to the norm || · ||  1 (R n ) , given by (2.8) (cf., e.g., Theorem 1.1 in Alliot and Amrouche 52 ).
In view of Lemma 2.5 of Kozono and Sohr, 53 the divergence operator div ∶  1 (R n ) n → L 2 (R n ) is surjective and has a bounded right inverse. Moreover, Remark 3.8(i) of Alliot and Amrouche 52 and Proposition 2.4(i) of Kozono and Sohr 53 imply that for n ≥ 3, the weighted Sobolev space  1 (R n ) can be also characterized as with equivalent norms.

Weighted Sobolev spaces on exterior Lipschitz domains
The weighted Sobolev space  1 (Ω − ) can be defined as in (2.7) with Ω − in place of R n . Therefore, is a Hilbert space with a norm given by (2.8) with Ω − in place of R n (see, e.g., Definition 1.1 in Alliot and Amrouche 37 ). The space −1 (Ω − ) is the dual of the space  1 (Ω − ).

Weighted Sobolev spaces on R n ∖ Ω
We also consider the weighted space This is a Hilbert space with the norm defined by which is equivalent to the norm (|| || 2

Rigid motion fields
Let  be the linear space of rigid body motion fields in R n , It is easy to see that dim  = n(n + 1)∕2; compare book Oleinik et al. 44 It is well known that if v ∈ H 1 (Ω ′ ) n and E(v) = 0 in a bounded domain Ω ′ , then v ∈ | Ω ′ (see, e.g., the proof of Theorem 2.5, chapter I in Oleinik et al. 44 ). This immediately implies that if Ω ′′ is R n or an exterior domain in R n and E(v) = 0 for v ∈  1 (Ω ′′ ) n ⊂ H 1 loc (Ω ′′ ) n , then v ∈ | Ω ′′ as well. Moreover, since v belongs to the space  1 (Ω ′′ ) n , which is embedded in L 2n n−2 (Ω ′′ ) n (see (2.10)), it follows that v = 0 in Ω ′′ .

The conormal derivative for the Stokes system with L ∞ viscosity tensor coefficient
As above, L is the divergence form of a second-order elliptic differential operator given by (1.7), and the coefficients A of the anisotropic tensor A = ( A ) 1≤ , ≤n are n × n matrix-valued functions in L ∞ (R n ) n×n , with bounded measurable, real-valued entries a i , satisfying the symmetry and ellipticity conditions (1.3) and (1.4). Moreover,  is the Stokes operator given by (1.9). Let = ( 1 , … , n ) ⊤ denote the outward unit normal to Ω + , which is defined a.e. on Ω.
In the special case when (u, ) ∈ C 1 (Ω ± ) n × C 0 (Ω ± ) and the coefficients a i are also continuous up to the boundary, the classical interior and exterior conormal derivatives (i.e., the boundary traction fields) for the Stokes system where f ∈ L 2 (Ω±) n , g ∈ L 2 (Ω±) are defined by the formula where T c± u are the conormal derivatives of u on Ω associated with the operator L and defined by . ‡ Note that for the isotropic case (1.10), the classical conormal derivatives t c± (u, ) reduce to the well-known formulas in the isotropic compressible case (cf., e.g., Appendix III, Part I, Section 1 in Temam 47 For the classical conormal derivatives defined by (2.20)-(2.22), the first Green formula holds and suggests the following definition of the generalized conormal derivative for the Stokes system with L ∞ viscosity tensor coefficient in the setting of weighted Sobolev spaces (cf., e.g., Lemma 4.3 in McLean, 48 25) † Here and in the sequel, the notation ± applies to the conormal derivatives from Ω±, respectively. ‡ Note that another type of conormal derivative, where E j (u) is replaced by its deviator, D (u) = E (u)− 1 n E mm (u) in the formulas like (2.22) and further on, has been considered in Fresdeda-Portillo and Mikhailov 56 for the isotropic case. Both types of conormal derivatives coincide for incompressible fluids.
In the sequel, we use the simplified notation t ± (u ± , ± ) for t ± (u ± , ± ; 0). LetE ± denote the operator of extension by zero outside Ω±. Thus, for a function v± from Ω± to R n , 33) and the jump of the corresponding formal or generalized conormal derivatives is denoted by In the special casef = 0, we use the notation Then, Lemma 1 implies the following assertion.
The following assertion is immediately implied by Lemma 2 and the symmetry conditions (1.3).

Lemma 3. Let conditions (1.2) and (1.3) hold. Let the pair (u, ) in
Then, for all w ∈  1 (R n ) n , the following formula holds:

Conormal derivative related to the adjoint Stokes operator
Let L be the divergence-type elliptic operator given by (1.7). Then, the formally adjoint L * of the operator L is defined by Note that the coefficients of the operator L * belong to L ∞ (Ω) n × n (cf. (1.2)) and satisfy the ellipticity condition (1.4) with the same constant c A . Moreover, the operator is the adjoint of the Stokes operator where f * ∈ L 2 (Ω±) n , then the corresponding classical conormal derivative is defined by  47) and the following variant of Lemma 3.

VARIATIONAL VOLUME AND LAYER POTENTIALS FOR THE ANISOTROPIC STOKES SYSTEM WITH L ∞ TENSOR COEFFICIENT
As in the previous sections, Ω + ⊂ R n , n ≥ 3, is a bounded Lipschitz domain with connected boundary Ω, and Ω − ∶= R n ∖Ω + . Recall that  is the Stokes operator defined in (1.9). In this section, we define the Newtonian and layer potentials for the Stokes system (1.8) by means of a variational approach.

Bilinear forms and weak solutions for the anisotropic Stokes system with
The subspace  1 div (R n ) n of  1 (R n ) n has also the characterization An important role in the forthcoming analysis is played by the following well-posedness result (see also Lemma 4.1 in Kohr et al. 39 .1) and (3.2), respectively. Then, for all given data F ∈  −1 (R n ) n and ∈ L 2 (R n ), the mixed variational formulation Proof. We intend to use Theorem 10, which requires the coercivity of the bilinear form a A;R n (·, ·) from  1 div (R n ) n ×  1 div (R n ) n to R. Indeed, the following Korn-type inequality for functions in  1 (R n ) n holds: This inequality is available, e.g., in Sayas and Selgas 33 , eq. (2.2) for n = 3. For arbitrary n ≥ 1, the Korn inequality is proved in Theorem 10.1 of McLean 48 for any function w ∈ (R n ) n . Hence, by the density of (R n ) n in  1 (R n ) n , this implies that inequality (3.6) is valid also in Then, the ellipticity condition (1.4), inequality (3.6), and equivalence of the seminorm ||∇(·) Inequality (3.7) shows that the bilinear form a A;R n (·, ·) ∶  1 div (R n ) n ×  1 div (R n ) n → R is coercive. The continuity of the operator ∇ ∶  1 (R n ) n → L 2 (R n ) n×n and the Hölder inequality imply that where  = n 4 ||A|| L ∞ (R n ) . Thus, the bilinear form a A;R n (·, ·) ∶  1 (R n ) n ×  1 (R n ) n → R is bounded. Moreover, the boundedness of the divergence operator div The isomorphism property of the divergence operator (cf. Proposition 2.1 in Alliot and Amrouche 52 and Lemma 2.5 in Kozono and Shor 53 ) implies that there exists a constant c 0 > 0 such that for any q ∈ L 2 (R n ) there exists v ∈  1 (R n ) n satisfying the equation −div v = q and the inequality ||v||  1 (R n ) n ≤ c 0 ||q|| L 2 (R n ) . Therefore, the following inequality holds for such v: This implies that the bounded bilinear form (see also Lemma 14(ii) and Proposition 2.4 in Sayas and Selgas 33 for n = 2, 3). Then, Theorem 10 with X =  1 (R n ) n ,  = L 2 (R n ), and V =  1 div (R n ) n implies that problem (3.4) is well-posed, as asserted.

Volume potential operators for the anisotropic Stokes system with L ∞ tensor coefficient
Recall that  is the anisotropic Stokes operator defined in (1.9).

Theorem 2. Let conditions (1.2)-(1.4) hold in R n .
Then, for each f ∈  −1 (R n ) n and g ∈ L 2 (R n ), the anisotropic Stokes system is well-posed, which means that (3.11) has a unique solution (u, ) ∈  1 (R n ) n × L 2 (R n ), and there exists a constant . (3.12) Proof. The dense embedding of the space (R n ) n in  1 (R n ) n shows that system (3.11) has the equivalent mixed variational formulation (3.4), with F = −f and = −g. Then, the well-posedness of the Stokes system (3.11) follows from Lemma 5.
Theorem 2 allows us to define the volume potential operators for the Stokes system with L ∞ coefficients and obtain their continuity as follows. (i) The Newtonian velocity and pressure potential operators, are defined as where (u f , f ) ∈  1 (R n ) n × L 2 (R n ) is the unique solution of problem (3.11) with f ∈  −1 (R n ) n and g = 0. (ii) The velocity and pressure compressibility potential operators, are defined as where (u g , g ) ∈  1 (R n ) n × L 2 (R n ) is the unique solution of problem (3.11) with g ∈ L 2 (R n ) and f = 0. Lemma 6. Operators (3.13) and (3.15) are linear and continuous and for any f ∈  −1 (R n ) n and g ∈ L 2 (R n ) n ,

The single-layer potential operator for the anisotropic Stokes system with L ∞ tensor coefficient
Recall that Ω + ⊂ R n (n ≥ 3) is a bounded Lipschitz domain with connected boundary Ω, Ω − ∶= R n ∖Ω + , the notation [·] is used for jumps (see formulas ( Proof. Transmission problem (3.17) has the following equivalent mixed variational formulation.
To show this equivalence, let us first assume that (u , ) ∈  1 (R n ∖ Ω) n × L 2 (R n ) satisfy transmission problem (3.17). Then, the first transmission condition in (3.17) implies the membership of u in  1 (R n ) n ; compare Lemma 16(ii). Moreover, formula (2.37) shows that the the pair (u , ) satisfies also the first equation in (3.19). The second equation in (3.19) follows from the second equation in the first line of (3.17).
Let us show the converse property. To this end, we assume that the pair (u , ) ∈  1 (R n ) n × L 2 (R n ) is a solution of the variational problem (3.19). By using the density of the space (R n ) n in  1 (R n ) n , and by considering in the first equation of (3.19) any v ∈ C ∞ (R n ) n with compact support in Ω±, we obtain the following variational equation: which leads to the first equation of the transmission problem (3.17). The second equation in (3.17) is an immediate consequence of the second equation in (3.19). On the other hand, the membership of u in  1 (R n ) n yields the first transmission condition in (3.17). In addition, formula (2.37) and the first equation in (3.19) show that Since the trace operator ∶  1 (R n ) n → H Thus, the transmission problem (3.17) has the equivalent mixed variational formulation (3.19), which can be written as where a A;R n and b R n are the bounded bilinear forms given by (3.1) and (3.2), and F ∈  −1 (R n ) n is defined as Then, by Lemma 5, the variational problem (3.19) is well-posed. Therefore, problem (3.17) has a unique solution (u , ) ∈  1 (R n ) n ×L 2 (R n ), which depends continuously on .
Theorem 3 allows to define the single-layer potentials for L ∞ coefficient Stokes system and to obtain their continuity.
are defined as 24) and the boundary operators, are defined as where (u , ) is the unique solution of the transmission problem (3.17) In addition, the following jump relations, that are similar to the case of the Stokes system with constant coefficients (see also Lemma 3.8 in Kohr et al., 43 (3.28)

The single-layer potential for the adjoint Stokes system
Recall that L * is the operator defined in (2.39), and t * is the conormal derivative operator for the adjoint Stokes system (see formula (2.47)). The next well-posedness result follows with an argument similar to that for Theorem 3 and is based on the Green formula (2.47).

29)
has a unique solution (v * , q * ) ∈  1 (R n ) n × L 2 (R n ), and there exists C * = C * ( Ω, c A , n) > 0 such that are defined as and the boundary operators, are defined as where (v * , q * ) is the unique solution of the transmission problem (3.29) in  1 (R n ∖ Ω) n × L 2 (R n ).

Lemma 9. Let conditions (1.2)-(1.4) hold. Then, the following formulas hold on :
Moreover, by the second formulas in (3.28) and (3.35), For a given operator T : X → Y, the set Ker {T ∶ X → Y } ∶= {x ∈ X ∶ T(x) = 0} is the null space of T. Let be the outward unit normal to Ω, which exists a.e. on Ω, and let span{ } ∶= {c ∶ c ∈ R}. Let also Next, we mention the main properties of the single-layer operator, similar to the ones provided in Lemma 3.12 in Kohr et al. 43 in the case of a strongly elliptic viscosity tensor coefficient (see also Lemma 4.9 in Kohr et al., 39 Proof. First, we note that the transmission problem (3.17) with the given datum = ∈ H − 1 2 ( Ω) n is well-posed. Let us show that the pair is the unique solution of this transmission problem. Indeed, (u , ) satisfies the equations and the first transmission condition in (3.17), and by formulas (2.25), (2.35), and (3.45), and by the divergence theorem, we obtain that Next, we apply formula (3.36) for the densities ∈ H − 1 2 ( Ω) n and ∈ H − 1 2 ( Ω) n and use the second relation in (3.47). Then, we obtain that ⟨ Ω , ⟩ Ω = ⟨ ,  * Ω ⟩ Ω = 0, and hence, (3.44) follows.

Isomorphism property of the single-layer operator
Next, we show the following invertibility property of the single-layer potential operator (cf. Lemma 3.13 in Kohr et al., 43 Theorem 10.5.3 in Mitrea and Wright, 5 Proposition 3.3 (d) in Bȃcuţȃ et al., 36 and Proposition 5.5 in Sayas and Selgas 33 in the case 1.10 with = 1 and = 0). 48) and the following operator is an isomorphism:
be the unique solution in  1 (R n ) n × L 2 (R n ) of the transmission problem (3.17) with the given datum 0 . In view of formula (2.37) and since u 0 = 0 on Ω, we obtain that In addition, since div u 0 = 0, we have E ii (u 0 ) = 0, and due to assumption (1.4), which implies that E(u 0 ) = 0 and hence u 0 = 0 in R n ; compare Section 2.2.4. Moreover, u 0 and 0 satisfy the Stokes equation in R n ∖ Ω and 0 belongs to L 2 (R n ). Thus, there exists c 0 ∈ R such that 0 = c 0 Ω + in R n . Then, formulas (2.25) and (2.35) and the divergence theorem yield that and accordingly that 0 = [t(u 0 , 0 )] = −c 0 . Taking into account (3.43), formula (3.48) follows.
(ii) Next, we use the notation [ ] for the classes of the space H − 1 2 ( Ω) n ∕span{ }. Thus, ⟦ ⟧ = + span{ }, with ∈ H − 1 2 ( Ω) n . We show that there exists a constant c = c( Ω, c A , n) > 0 such that the single-layer potential operator satisfies the coercivity inequality where u = V Ω and =  s Ω . Moreover, the trace operator ∶  1 div (R n ) n → H

The double-layer potential operator for the anisotropic Stokes system with L ∞ viscosity tensor coefficient
Note that if u ∈ L 2,loc (R n ) n is such that u| Ω + ∈ H 1 (Ω + ) n , u| Ω − ∈  1 (Ω − ) n , then, due to Definition (2.16), u ∈  1 (R n ∖ Ω) n and can be endowed with the norm ||u|| 2  1 (R n ∖ Ω) n ∶= ||u|| 2 H 1 (Ω + ) n + ||u|| 2  1 (Ω − ) n that is equivalent to the norm (2.17). By following a similar approach to that used to define the Stokes single-layer potentials, we now show the well-posedness of a transmission problem that allows us to define the L ∞ -coefficient Stokes double-layer potentials with the densities in the space H 1 2 ( Ω) n , n ≥ 3 (cf. Theorem 3.14 in Kohr et al. 43 for the Stokes system with strongly elliptic tensor coefficient and Propositions 6.1 and 7.1 in Sayas and Selgas 33 for the isotropic case 1.10 with = 1, = 0, and n = 2, 3). Let conditions (1.2)-(1.4) hold on R n . Then, for any ∈ H 1 2 ( Ω) n , the transmission problem,

Theorem 5.
has a unique solution (u , ) ∈  1 (R n ∖ Ω) n × L 2 (R n ), and there exists C = C( Ω, c A , n) > 0 such that Proof. First, we show the uniqueness. Let (u 0 , 0 ) ∈  1 (R n ∖ Ω) n × L 2 (R n ) be a solution of the homogeneous version of problem (3.56). Therefore, the couple (u 0 , 0 ) is a solution of the homogeneous version of the transmission problem (3.17), which, in view of Theorem 3, has only the trivial solution.
Next, we show that the transmission problem (3.56) has the following equivalent variational formulation. (3.58) Indeed, if (u , ) ∈  1 (R n ∖ Ω) n × L 2 (R n ) satisfies transmission problem (3.56), then the Green formula (2.37) yields the first equation of problem (3.58). The second equation of (3.58) is the distributional form of the second equation of (3.56). Conversely, assume that (u , ) ∈  1 (R n ∖ Ω) n × L 2 (R n ) satisfies the variational problem (3.58). Then, from the first equation of (3.58), we deduce that which is the distributional form of the first equation in (3.56). The second equation of (3.56) follows from the second equation of (3.58). In addition, the first equation of (3.58) and the Green formula (2.37) applied to the pair (u , ) yield that Therefore, v ∶= u − w satisfies the condition [ v ] = 0, and hence, by Lemma 16 can be extended to v ∈  1 (R n ) n . In addition, (3.58) reduces to the following variational problem: Then, Lemma 5 implies that the variational problem (3.62) has a unique solution (v , ) ∈  1 (R n ) n × L 2 (R n ). Hence, the pair (u , ) = (w +v , ) is a solution of the variational problem (3.58) in the space  1 (R n ∖ Ω) n ×L 2 (R n ) and depends continuously on . The equivalence between problems (3.56) and (3.58) show that (u , ) is the unique solution of the transmission problem (3.56).
Theorem 5 suggests the following definition of the double-layer potential operator for the anisotropic Stokes system (1.8) in the case n ≥ 3 (cf. Sayas and Selgas 33 , p.77 for the constant-coefficient Stokes system in R 3 , formula (4.5) and Lemma 4.6 in Barton 29 for general strongly elliptic differential operators, and Definition 3.15 in Kohr et al. 43 for the Stokes system with L ∞ strongly elliptic viscosity coefficient). (3.66) and the boundary operators, (3.67) are defined as K Ω ∶= 1 2

Definition 5. Let conditions (1.2)-(1.4) hold. Then, the double-layer velocity and pressure potential operators,
where (u , ) is the unique solution of the transmission problem (3.56) Moreover, the well-posedness of the transmission problem (3.56) and Definition 5 lead to the next result (see also formulas (10.81) and (10.82) in Mitrea and Wright 5 and Propositions 6.2 and 6.3 in Sayas and Selgas 33 for the constant-coefficient Stokes system in R 3 , and Lemma 5.8 in Barton 29 for strongly elliptic operators). Let conditions (1.2)-(1.4) are satisfied. Then, the following assertions hold.

Lemma 12.
(i) Operators (3.65) and (3.67) are linear and continuous and for any ∈ H − 1 2 ( Ω) n , (ii) For any ∈ H 1 2 ( Ω) n , the following jump formulas hold on Ω (3.34) is the transpose of the double-layer operator K Ω ∶ H 1 2 ( Ω) n → H 1 2 ( Ω) n defined in (3.68), that is, Proof. The continuity of operators (3.65) and (3.67) follows from the well-posedness of transmission problem (3.56) and Definition 5. By invoking again Definition 5 and the transmission conditions in (3.56), we obtain jump formulas (3.70). Next, we show equality (3.71), by using an argument similar to that in the proof of Proposition 6.7 in Sayas and Selgas 33 for the constant-coefficient Stokes system. Let ∈ H 1 2 ( Ω) n be given, and let (u , ) = be the unique solution of the problem (3.56) with datum . Let also * ∈ H − 1 2 ( Ω) n and (v * , q * ) = (V * Ω * ,  s * Ω * ) ∈  1 (R n ) n × L 2 (R n ) be the solution of the problem (3.29) with datum * , that is, the single-layer velocity and pressure potentials with density * for the adjoint Stokes system (see Definition 4). Then, by formulas (2.37) and (3.69), Moreover, the Green formula (2.47) for the adjoint Stokes system and equality (3.72) yield that (3.73) Therefore, we obtain the equality Then, the second formula (3.35), the first formula (3.70), and formula (3.74) lead to the equality and hence to equality (3.71), as asserted.

Invertibility of the operator D Ω
Let  be the set of rigid body motion fields in R n , see (2.18), and let It is easy to see that E(r) = 0, div r = 0 ∀ r ∈ . (3.78) Next, we show the isomorphism property of the operator D Ω defined in (3.69) (cf. Lemma 3.17 in Kohr et al. 43 for a different structure of the kernel and range of the similar operator when A is an L ∞ strongly elliptic viscosity tensor coefficient and Propositions 6.4 and 6.5 in Sayas and Selgas 33 for the Stokes system with constant coefficients). (3.80) and the following operator is an isomorphism,
(i) First, we show formula (3.79). Let us assume that ∈ H 1 2 ( Ω) n satisfies the equation D Ω = 0 on Ω. Let u ∶= W Ω and ∶=  d Ω . Since div u = 0 in Ω±, we have E ii (u ) = 0 implying that assumption (1.4) is applicable for E i (u ). According to Lemma 1, the jump relations (3.70) and (3.69), and assumption (1.4), we obtain that and accordingly E(u ) = 0 in Ω±. Hence, by the statement in Section 2.2.4, there exist a constant b ∈ R n and an antisymmetric matrix B ∈ R n×n such that u = b + Bx in Ω + , while u = 0 in Ω − . Then, by using again the jump relations (3.70), we obtain that = −(b + Bx)| Ω . This relation shows that Now, let r ∈  and let u r and r be the fields given by By (3.78), E(u r ) = 0 and div u r = 0 in R n ∖ Ω, and hence, in view of Lemma 1, which show that t ± (u r , r ) = 0, and accordingly that [t(u r , r )] = 0 on Ω. Moreover, we have that [ u r ] = −r| Ω on Ω. Consequently, the pair (u r , r ) belongs to  1 (R n ∖ Ω) n ×L 2 (R n ) and satisfies the transmission problem (3.56) with given boundary datum r| Ω ∈ H 1 2 ( Ω) n . Then, Definition 5 yields that W Ω (r| Ω ) = u r and  d Ω (r| Ω ) = 0 in R n ∖ Ω, and by formula (3.69), we obtain that D Ω (r| Ω ) = 0 on Ω. Therefore, implying formula (3.80). (ii) To prove that operator (3.81) is an isomorphism, we show that there exists a constant  = ( Ω, c A , n) > 0 such that (cf. Sayas and Selgas 33 Proposition 6.5 in the constant-coefficient Stokes system). Indeed, by applying Lemma 1 to the  ( Ω) n , and using the jump relations (3.70) and condition (1.4), we obtain the inequality In addition, the continuity of the trace operators ± ∶  1 (Ω ± ) n → H 1 2 ( Ω) n and the jump formulas (3.70) imply that there exists a constant  1 =  1 ( Ω, c A , n) > 0 such that } be a basis of the n(n + 1)/2-dimensional space . Then, the formula defines a norm on the space  1 (R n ∖ Ω) n , which is equivalent to the norm || · ||  1 (R n ∖ Ω) n (see Lemma 18, cf. also Sayas and Selgas 33 , p.78 for n = 3). Therefore, there exists a constant  2 > 0 such that Now, by considering w = u in (3.91) and by using the jump formulas (3.70), and the assumption that ∈ H 1 2  ( Ω) n , as well as inequality (3.92), we obtain that

The third Green identities for the anisotropic Stokes system
Next, we prove the representation formulas (the third Green identities) for solutions of the anisotropic Stokes system with L ∞ tensor coefficient (cf. Proposition 6.8 in Sayas and Selgas 33 for the homogeneous Stokes system in case (1.10) with = 1, = 0, and n = 3 and Theorem 6.10 in McLean 48 for the strongly elliptic systems with smooth coefficients). They can be employed, for example, for reduction of the boundary and transmission problems to direct boundary equations, similar to the classical direct boundary integral equation approach, see, for example, Costabel, 1 McLean, 48 (1.9). Let u + ∈ H 1 (Ω + ) n , u − ∈  1 (Ω − ) n and ± ∈ L 2 (Ω±) satisfy the Stokes system (u ± , ± ) =f ± | Ω ± , div u ± = g ± in Ω ± (3.98) for somef + ∈H −1 (Ω + ) n ,f − ∈ −1 (Ω − ) n , g + ∈ L 2 (Ω + ), g − ∈ L 2 (Ω − ). Letf ∶=f + +f − , g ∶=E + g + +E − g − . Then, the following representations in terms of jumps hold: Moreover, the following single-side representations also hold: Proof. In view of the assumptions on u±, ±, andf ± , we have the inclusions By definitions of the potentials and according to Lemmas 6,7,and 12(i), the pair (v, q) belongs to the space  1 (R n ∖ Ω) n × L 2 (R n ) and satisfies the Stokes system Then, by formulas (3.27) and (3.70), [ v] = on Ω. (3.105) Let r Ω ± be restriction operators to Ω±, that is, r Ω ± g ∶= g| Ω ± . By Definition 1, the generalized conormal derivative is linear with respect to the triple of its arguments, implying that (3.106) By formulas (3.28) and (3.69), we obtain On the other hand, from (2.25), we have for any w ∈ H 1 2 ( Ω) n that The last equality in (3.108) follows since ( R nf + R n g,  R nf + 0 R n g) =f in R n . Combining (3.106)-(3.108), we obtain that the couple (v, q) satisfies the transmission condition

Exterior Dirichlet problem for the anisotropic Stokes system in the compressible case
Let us consider the following Dirichlet problem for the anisotropic Stokes system with L ∞ coefficients: where  is the Stokes operator defined in (1.9) and the given data (f, g, ) belong to  −1 (Ω − ) n × L 2 (Ω − ) × H wheref is an extension of f to an element of −1 (Ω n − ) ⊂  −1 (R n ) n . In addition, there exists a constant C = C( Ω, c A , n) > 0 such that Proof. Let f ∈  −1 (Ω − ) n and g ∈ L 2 (Ω − ). Then, Theorem 3.2 and Definition 3.3 imply that and div  R nf = 0, div  R nE − g =E − g in R n . Hence, both potentials are divergence free vector fields in Ω + and the divergence theorem in Ω + implies that From inclusions (4.5), we have which, together with (4.6), implies that −  R nf, −  R nE − g ∈ H 1 2 ( Ω) n . Then, by the assumption ∈ H 1 2 ( Ω) n , we 2 ( Ω) n , and, in view of Lemma 11,  −1 is a well-defined element of the space H − 1 2 ( Ω) n ∕span{ }. Moreover, Lemmas 6,7,8,and 11 imply that (u, ) represented by formulas (4.2) and (4.3) solve the exterior Dirichlet problem (4.1) in  1 (Ω − ) n × L 2 (Ω − ), and the continuity of the operators involved in these formulas yields inequality (4.4).

Exterior Neumann problem for the anisotropic Stokes system
The Neumann problem for the constant coefficient Stokes system in an exterior Lipschitz domain in R n , with boundary datum in L p spaces, has been studied in Theorem 9.2.6 of Mitrea and Wright 5 by a potential approach (see also Theorem 10.6.4 in Mitrea and Wright 5 for the Neumann problem for the same system in a bounded Lipschitz domain). Next, we consider the following exterior Neumann problem for the L ∞ coefficient Stokes system: ∈  ⟂ Ω , then problem (4.9) has a unique solution (u, ) ∈  1 (Ω − ) n × L 2 (Ω − ), given by Moreover, there exists a constant C = C(Ω − , c A , n) > 0 such that (4.11) Proof. Lemmas 12 and 13 imply that (u, ) represented by (4.10) solve problem (4.9) and the operators involved in (4.10) are continuous, which implies inequality (4.11).

Abstract mixed variational formulations
A major role in our analysis of mixed variational formulations is played by the following well-posedness result by Babuska 61  Theorem 10. Let X and  be two real Hilbert spaces. Let a(·, ·) ∶ X × X → R and b(·, ·) ∶ X ×  → R be bounded bilinear forms. Let f ∈ X ′ and g ∈  ′ . Let V be the subspace of X defined by Assume that a(·, ·) ∶ V × V → R is coercive, which means that there exists a constant C a > 0 such that

2)
and that b(·, ·) ∶ X ×  → R satisfies the Babuska-Brezzi condition with some constant C b > 0. Then, the mixed variational formulation, has a unique solution (u, p) ∈ X ×  and where ||a|| is the norm of the bilinear form a(· , ·).
We need also the following extension of the Babuška-Brezzi result (see Theorem 4.2 in Amrouche and Seloula; 65 see also Lemma A.40 in Ern and Guermond 63 ).

Lemma 14.
Let X and  be reflexive Banach spaces. Let b(·, ·) ∶ X ×  → R be a bounded bilinear form. Let B ∶ X →  ′ and B * ∶  → X ′ be the linear bounded operator and its transpose operator defined by where ⟨·, ·⟩ ∶= X ′ ⟨·, ·⟩ X denotes the duality pairing between the dual spaces X ′ and X. The duality pairing between the spaces  ′ and  is also denoted by ⟨· , ·⟩.
Then, the following assertions are equivalent: (i) There exists a constant C b > 0 such that b(· , ·) satisfies the inf-sup condition (5.3).
(ii) The operator B ∶ X∕V →  ′ is an isomorphism and iii) The operator B * ∶  → V ⟂ is an isomorphism and
The Stokes system is elliptic in the sense of Agmon-Douglis-Nirenberg at x ∈ R n if (x, ) is defined and nonsingular for any ∈ R n ∖{0} (see, e.g., Definition 6.2.3 in Hsiao and Wendland 4 ). This property is well known for the Stokes system in the isotropic case (1.10) with = 1 and = 0 (cf., e.g., Hsiao and Wendland 4 , p.329). Next, we show that this ellipticity property remains valid also in the more general anisotropic case. is nonsingular as well. Let x ∈ R n be such that the coefficients a (x) are well defined and finite and the ellipticity condition (1.4) holds. In order to show that̃(x, ) is nonsingular for any ∈ R n ∖{0}, we use Theorem 10. To this end, for a fixed ∈ R n ∖{0}, we consider the bilinear forms a 0 ∶ R n × R n → R and b 0 ∶ R n × R → R, b 0 (v,q) ∶= −vq ∀v ∈ R n ,q ∈ R, (5.13) as well as the closed subspace V of R n given by It is immediate that these bilinear forms are bounded, as they satisfy the estimates: |a 0 (û,v)| ≤ ||A|| L ∞ (R n ) | | 2 |û||v|, |b 0 (v,q)| ≤ | ||v||q| ∀û,v ∈ R n , ∀q ∈ R.

Extension result in the weighted
Proof.
To prove that the function u is unique, let us assume that there are two such functions, u 1 and u 2 . Then, u 0 : = u 1 − u 2 belongs to  1 (R n ) and u 0 | Ω ± = 0. Thus, u 0 ∈ H 1 (R n ) ⊂ L 2 (R n ) and its support is a subset of Ω. Hence, u 0 = 0 in R n in the sense of Lebesgue classes (cf. also Theorem 2.10(i) in Mikhailov 54 ). (iii) Let u ∈  1 (R n ). Consequently, u ∈ H 1 loc (R n ), and then, + u = − u, that is, [ u] = 0.

Equivalent norms in the weighted Sobolev space  1 (R n ∖ Ω) n
We will further employ the following assertion concerning the equivalence of norms in Banach spaces (cf. Lemma 11.1 in Tartar 66 ).
The following result for n = 3 is implied by Proposition 2.7(a) in Sayas and Selgas, 33 and its proof is based on the Korn inequalities (see, e.g., Theorems 10.1 and 10.2 in McLean 48 ) and Lemma 17. The result for n > 3 follows with the same arguments.